A new characterization of the CR sphere and the sharp eigenvalue estimate for the Kohn Laplacian

In Riemannian geometry, the well-known Lichnerowicz–Obata theorem gives a sharp estimate and a characterization of equality case (rigidity theorem) for first positive eigenvalue of Laplacian. In CR geometry, analogous problems are more delicate due to the presence of the torsion of Tanaka–Webster connection. Estimates and rigidity theorems for the first positive eigenvalue of the sub-Laplacian have been studied by many authors (see [11] and the reference therein), e.g., Li and Wang proved an Obata-type theorem for sub-Laplacian without any additional assumption on the torsion [11].In [6], Chanillo, Chiu and Yang gave a sharp eigenvalue estimate for the Kohn Laplacian on three-dimensional manifolds. This poses the question whether a version of rigidity theorem holds in this case. A partial answer was given by Chang and Wu in 2012. They generalized the eigenvalue estimate to general dimension and proved, under additional assumptions on the torsion, that the equality holds only when the manifold is the CR sphere. In the present paper, we completely resolve the rigidity question for manifolds of dimension at least five by establishing a new characterization of CR sphere in terms of the existence of a (non-trivial) function satisfying a certain overdetermined system. The result holds without any assumption on the pseudohermitian torsion.